image and video coding and processing lecture 2 basic filtering l.
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Image (and Video) Coding and Processing Lecture 2: Basic Filtering. Wade Trappe. Lecture Overview. Today’s lecture will focus on: Review of 1-D Signals Multidimensional signals Fourier analysis Multidimensional Z-transforms Multidimensional Filters. 1-D Discrete Time Signals.

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lecture overview
Lecture Overview
  • Today’s lecture will focus on:
    • Review of 1-D Signals
    • Multidimensional signals
    • Fourier analysis
    • Multidimensional Z-transforms
    • Multidimensional Filters
1 d discrete time signals
1-D Discrete Time Signals
  • A one-dimensional discrete time signal is a function x(n)
  • The Z-transform of x(n) is given by
  • The Z-transform is not guaranteed to exist because the summation may not converge for arbitrary values of z.
  • The region where the summation converges is the Region of Convergence
  • Example: If U(n) is the unit step sequence, an x(n)=anU(n), then
1 d discrete time signals pg 2
1-D Discrete Time Signals, pg. 2
  • If the ROC includes the unit circle, then there is a discrete Fourier transform (found by evaluating at z=ejw):
  • The inverse transform is given by:
  • Observe: The DFT is defined in terms of radians! It is therefore periodic with period 2p!
  • Parseval/Plancherel Relationship:
1 d discrete time signals pg 3
1-D Discrete Time Signals, pg. 3
  • Discrete time linear, time-invariant systems are characterized by the impulse response h(n), which define the relationship between input x(n) and output y(n)
  • This is convolution, and is expressed in the transform domain as:
  • Causality: A discrete-time system is causal if the output at time n does not depend on any future values of the input sequence. This requires that
1 d filters
1-D Filters
  • The impulse response for a system is also called the system’s transfer function.
  • In general, transfer functions are of the form
  • A system is a finite impulse response (FIR) system if H(z)=A(z), i.e. we can remove the denominator B(z)
    • That is, the impulse response has a finite amount of terms.
  • An infinite impulse response system is one where H(z) has an infinite amount of non-zero terms.
  • Example of an IIR system:
1 d filters pg 2
1-D Filters, pg. 2
  • A discrete-time system is said to be bounded input bounded output (BIBO stable), if every input sequence that is bounded produces an output sequence that is bounded.
  • For LTI systems, BIBO stability is equivalent to
  • Stability in terms of the poles of H(z):
    • If H(z) is rational, and h(n) is causal, then stability is equivalent to all of the poles of H(z) lying inside of the unit circle
sampling from continuity to discrete
Sampling: From Continuity to Discrete
  • The real world is a world of continuous (analog) signals, whether it is sound or light.
  • To process signals we will need “sampled” discrete-time signals
  • Analog signals xa(t) have Fourier transform pairs
  • Let us define the sampled function x(n)=xa(nT). The Fourier transforms are related as:
  • (Note: This is a good, little homework problem… will be assigned!)
sampling from continuity to discrete9






Sampling: From Continuity to Discrete
  • The effect of the sampling in the frequency domain is essentially
    • Duplication of Xa(W) at intervals of 2p/T
    • Addition of these “copies”
  • Pictorially, we have something like the following:
  • Note: If the shifted copies overlap, then its “impossible” to recover the original signal from X(w).

1/T Xa(W)

Shifted Copies


sampling from continuity to discrete pg 2
Sampling: From Continuity to Discrete, pg. 2
  • Aliasing occurs when there is overlap between the shifted copies
  • To prevent aliasing, and ensure recoverability, we can apply an “anti-aliasing” filter to ensure there is no overlap.
  • The overlap-free condition amounts to ensuring that
  • If , then we say that xa(t) is W-bandlimited.
  • As a consequence of the overlap-free condition, if we sample at a rate at least W, then we can avoid aliasing.
  • This is, essentially, Shannon’s sampling theorem.
multidimensional signals
Multidimensional signals
  • A D-dimensional signal xa(t0,t1,…,tD-1) is a function of D real variables.
  • We will often denote this as xa(t), where the bold-faced t denotes the column vector t=[t0, t1, …, tD-1]T.
  • The subscript “a” is just used to denote the analog signal. Later, we shall use the subscript “s” to denote the sampled signal, or no subscript at all.
  • The Fourier transform of xa(t) is defined by
multidimensional signals pg 2
Multidimensional signals, pg. 2
  • The Fourier transform is thus a scalar function of D variables.
  • The Fourier transform is (in general) complex!
  • The Inverse Fourier transform of Xa(W) is defined by
  • Define the column vector of frequencies
  • We get these relationships

Note the difference that D-dimensions introduces compared to 1-d Fourier Transform!

example 2d fourier transform
Example 2D Fourier Transform

Note: Ringing artifacts, just like 1-D case when we Fourier Transform a square wave

Image example from Gonzalez-Woods 2/e online slides.

bandlimited signals





Bandlimited Signals
  • The notion of a bandlimited multidimensional signal is a straight-forward extension of the one-dimensional case:
    • xa(t) is bandlimited if Xa(W) is zero everywhere except over a region with finite area.


Not Bandlimited

multidimensional sampled signals
Multidimensional Sampled Signals
  • We will use n=[n0,n1,…,nD-1]T to denote an arbitrary D-dimensional vector of integer values
  • A signal x(n) is just a function of D integer values
  • The Fourier transform of x(n) and the inverse transform are given by
  • Key point: X(w) is periodic in each variable wi with period 2p
multidimensional z transform
Multidimensional Z transform
  • The Z transform of x(n) is
  • Plugging in gives X(w).
  • We will often use the notation
  • This notation will be useful later as it allows us to represent things in a way similar to the 1-dimensional Z-transform
properties of fourier and z transforms
Properties of Fourier and Z transforms
  • Linearity
  • Shift:

Hence, the multidimensional Z-transform is analogous to the one-dimensional delay operator

  • Convolution:
multidimensional filters
Multidimensional Filters
  • The basic scenario for multidimensional digital filters is:
  • Convolution:
  • Here, the transfer function is
  • If x(n) has finite support, then y(n) will generally have larger support than x(n)




multidimensional filter response







Multidimensional Filter Response
  • Just as in 1-D, the filter H can be characterized in terms of its frequency response.
  • In this case, the frequency response is

Rectangular Lowpass

Diamond Lowpass

Circular Lowpass

multidimensional filters20





Multidimensional Filters
  • Multidimensional filters can be built by applying 1-D filters to each dimension separately
  • These types of filters are separable.
  • A separable filter is one for which the frequency response can be represented as:

Rectangular Lowpass

Not Separable

2 d convolution by hand
2-D Convolution, by hand…
  • Rotate the impulse response array h(  ,  ) around the original by 180 degree
  • Shift by (m, n) and overlay on the input array x(m’,n’)
  • Sum up the element-wise product of the above two arrays
  • The result is the output value at location (m, n)

From Jain’s book Example 2.1

for next time
For Next Time…
  • Next time we will focus on multidimensional sampling.
    • This lecture will be a blackboard/whiteboard style lecture.
  • To prepare, read paper provided on website, and the discussion on lattices in the textbook